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Solution of static reduced decoupling problem for linear
systems
Jean François Lafay
To cite this version:
Jean François Lafay. Solution of static reduced decoupling problem for linear systems. 2013.
�hal-00948657�
Solution of Static Reduced Decoupling
Problem for Linear Systems
Jean-Fran¸cois Lafay∗ In memoriam A.N. Herrera H.
∗Institut de Recherche en Communications et Cybern´etique de Nantes,
IRCCyN (UMR CNRS 6597), Ecole Centrale de Nantes, France.
Abstract: We propose a structural solution by non-regular static state feedback to the diagonal, or row by row decoupling problem for linear systems. Without being completely general, this solution concerns the reduced Morgan’s problem, that is we do not want it increases the essential orders of the open loop system. The solution that we propose relies on properties between some partial infinite structures extracted from well chosen interactors and controllability indices of a specific shifted system. To my knowledge, there is at this date only very partial solutions to this problem deemed structurally very difficult.
Keywords: linear systems, non regular control, state feedbacks, diagonal decoupling, structural properties.
1. INTRODUCTION
Diagonal decoupling by a static state feedback, u = F x + Gv, or Morgan’s Problem, is a very difficult control problem when G is not invertible. Such non regular feedbacks correspond to a loss of inputs for the controlled system. They can transform all invariant structures underlying the regular solutions to control problems. The challenge is to find new structures that will allow us to solve Morgan’s Problem, knowing that non regular static solutions can exist even when there is none with regular laws. The non-regular diagonal decoupling by dynamic state feedback is much simpler as by static state feedback. It was solved by Dion and Commault [1988] due to new invariants, the essential orders. Non regular decoupling always comes down to a problem of increasing structures. The relative simplicity of the dynamic case comes from the fact that the essential orders are the minimum infinite structure to achieve for decoupling and that it is always possible to get it if there is a solution. The reason is that, to make this increase, dynamic feedback uses dynamics (integrators) that are external to the system. This is no more the case for static feedbacks that must only use internal dynamics. We consider here the Static Reduced Morgan’s Problem (SRMP), say the static decoupling without increasing the essential orders: it provides insight into the complex mechanisms of structural changes by non-regular controls and, from the practical point of vue, this restricted problem is not whitout interest because it is necessary that a dynamic solution exists, (therefore maintaining the essential orders), to hope to find a static one. There are so far only very partial results for the SRMP: when it is suffisant to increase only one element of the infinite structure (Herrera H and Lafay [1993]), or for trivial internal structures, (Zagalak et al. [1998]),(Lafay [2013]). The specific difficulties of SRMP are of two kinds: firstly, although their number is the same and is given
by the necessary and sufficient condition of the dynamic problem, the increases of infinite structure for solving SRMP depend on the order of the outputs of the system, while knowing that the sum of the sizes of these increases does not depend of this order. This first lock has been lift in Lafay [2013], where it is proved that there is a unique ”minimal list of decoupling indices” such that if SRMP has a static solution, one solution exist with this minimal indices. Secondly we must take into account internal couplings which are unobservable from the outputs to be decoupled. For that, we develop a non trivial general formulation of SRMP inspired by (Herrera H and Lafay [1993]). The general solution of SRMP relies on the controllability of a well defined shifted system .
2. NOTATIONS AND BACKGROUND
2.1 Notations
Let Σ(C, A, B), denoted shortly Σ, a linear system whose state is supposed to be measured or reconstructible:
Σ ˙x(t) = Ax(t) + Bu(t)
y(t) = Cx(t). (1) x ∈ X ⊂ Rn, u ∈ U ⊂ Rm and y ∈ Y ⊂ Rp are outputs
to be controlled. We suppose B monic and C epic. Σ is controllable, right invertible and without finite zeros. This last assumption implies that V∗=R∗, V∗ and R∗ being the supremal (A − B)invariant and the supremal controllability subspaces in the kernel of C(finite zeros concern only internal stability, which is not adressed here). A set of p elements is noted {•}p. Polynomial and rational
objects in variable s are respectively noted •[s] and •(s). We note ∂p[s] the degree of a polynomial and ∂ciM [s] the
highest degree of the ith column of matrix M[s]. Derivative of order n of function f (t) is noted f(n)(t). In represents
the identity matrix of order n and diag{ai}p the p × p
Definition 1. The interactor of Σ,(Wolovich and Falb [1976]), is the unique p × p triangular and non-singular polynomial matrix Φ[s] = [ϕi,j[s]] such that there exists
a biproper m × m (non unique) matrix B(s) satisfying : T (s) = C(sIn− A)−1B =Φ−10 B(s) where:
• ϕi,i= sdi, i = 1, ..., p, di being positive integers
• ϕi,j is zero or ϕi,j/sdi is divisible by s, ∀j > i.
The interactor is invariant under the action of the group (T, F, G), where T and G are changing of bases on X and U , and F is a state feedback. Without any loss of generality, we can suppose that it is row reduced (r.r.), (Lafay et al. [1992]).
Four lists of integers characterize partially the structure of Σ, (see Morse [1973],Commault et al. [1986]):
- {ci}m: controllability indices of (A, B) ,
- {n0i}p: I4 Morse’s list of the orders of the zeroes at
infinity. If the interactor of Σ is r.r, di= n0i, i = 1, ..., p.
- {σj}m−p: I2 Morse’s list providing the structure of R∗,
- {nie}p: the p essential orders of the outputs yi(t):
nie= ∂ciΦ[s], i = 1, ..., p, .
These lists are invariant under the action of the group (T, F, G, Π), where Π is a permutation of the outputs of Σ and G is invertible (Herrera H et al. [1997]).
An extended system Σe(Ce, A, B) is defined in Herrera H
et al. [1997] by adding m − p ficticious outputs, to reflects the structure of R∗ taking:
Ce= C (0)p×Pm−p j=1σj (0)(m−p)×Pp i=1n 0 i diag[1 0 ... 0]1×σj m−p (2).
Proposition 1. The m × m interactor of Σe, called ”ex-tended interactor of Σ”, has the structure:
Φe[s] =Φ e 1[s] (0) Φe2[s] Φe3[s] =ϕe i,j[s] , where (3) • Φe[s] is row reduced • Φe
1[s] is the p × p interactor of Σ, then ϕei,i= sn
0 i, i = 1, ...p, • Φe 3[s] = diag {sσ1, ..., sσm−p}, and • Polynomialsϕe i,j[s] = P h j=lαjs j of Φe 2[s] are zero or verify: n0j+ 1 ≤ l and h ≤ σi− 1 . Factorize Φ[s] as Φ[s] = W (s)diag{snie} p. W (s) is a
proper matrix, of rank at infinity k, called ”proper part of Φ[s].” (Dion and Commault [1988]) :
Proposition 2. There exists output’s permutation(s) Π such that the p × p interactor ΦΠof ΣΠ(ΠC, A, B) has the
structure: ΦΠ=ΦΠ,1 [s] (0) ΦΠ,2[s] ΦΠ,3[s] =ϕΠi,j[s] , where (4)
• The k × k polynomial matrix ΦΠ,3[s] is the diagonal
matrix = diag{snje}
k, the k integers nje being the
essential orders of the k last outputs of ΣΠ,
• Non zero elements of the infinite structure of the proper part of ΦΠ are given by the list {δi}p−k =
{nie− fi}p−k, where fi = ∂ϕΠi,i[s] for i = 1, ..., p−k.
• {δi}p−k is the list of decoupling indices of Σπ,
• There are always, among permutations Π, some per-mutations Πmsuch that the list {δi}p−kis the unique
minimal list {∆i}p−k of decoupling indices of Σ,
• Note that in general ΦΠ(and ΦΠm) are no longer r.r.
The minimal list of decoupling indices, is defined in (Lafay [2013]) using the notion of ”minor” list:
Definition 2. Let two lists of integers {δi}k1 and {γi}k2
such thatPk1
i=1δi=
Pk2
i=1γi. Note {ˆδi}supδiand {ˆγi}supγi
their dual lists. {δi}k1 ”minore” the list {γi}k2 if, for
i = 1, ..,sup(supj(δj), supj(γj)), we have: i X j=1 ˆ δj ≤ i X j=1 ˆ γj. (5) .
This list {∆i}p−k and an algorithmic procedure to obtain
it are presented in Appendix A in a better way as in Lafay [2013] .
2.2 Impact of a permutation Π on the extended interactor
We consider the extended interactors of Σ(C, A, B) and ΣΠ(ΠC, A, B). We note Φ[s] = [ϕi,j], and Φe[s] is
structured as in (3).
Property 1. Let a right invertible system Σ. Let Π any permutation matrix acting on the outputs of Σ. The extended interactor of ΣΠis given by
ΦeΠ[s] = Φe Π,1[s] (0) Φe Π,2[s] Φ e Π,3[s] , where : (6) • Φe Π,1[s] = ΦΠ[s] = B1(s)Φ[s]Π−1, where ΦΠ[s] is the
interactor of ΣΠand B1(s) is a biproper p × p matrix,
• Φe
Π,2[s] = B2(s)Φ2e[s]Π−1, where B2(s) is a proper
(m − p) × p matrix, which can add polynomials to whose of Φe
2[s] but, in each column i, these
polyno-mials are all of degree less than or equal to nie.
• Φe
Π,3[s] = Φe3[s]
The proof is based on the properties of biproper transfor-mations. It is not given here for sake of shortness. Remark 1. Let us consider equations (3) and (6). Suppose that, for i = 1, ...p − k, one keeps in the ith columns of Φe2[s] and ΦeΠ,2[s] only monomials of degree higher than nie: then, the so ”truncated” matrices Φe,t2 [s] and Φ
e,t Π,2[s]
satisfy
Φe,tΠ,2[s] = Φe,t2 [s]Π−1 (7) 2.3 On the model of the system
All the transformations involved to obtain interactors or extended interactors correspond to left biproper trans-formations which can be globally realized by a regular state feedback (Fm, Gm), (Gm invertible), on Σ or Σe,
(Hautus and Heymann [1978]). Then they did not affect the static decouplability of Σ and we can suppose that the transfer matrix Tm(s) of Σm is Φm−1(s). We note
Σm(Cm, Am, Bm), where Am= A+BFm, Bm= BGmand
Cm= ΠmC, Fmand Gmbeing a regular state feedback .
In all what follows it will be assumed, unless ex-plicitly mentioned, that Σ is already structured as Σm,
(Proposition 2), with {∆i}p−k= {δi}p−k. To summarize,
the extended interactor Σe m is : Φem[s] =ϕe mi,j[s] = Φe m1,1 (0) (0) Φem1,2 Φem1,3 (0) Φe m2,1 Φem2,2 Φem2,3 (8)
where the p × p interactor of Σmis:
Φm[s] = Φe m1,1 (0) Φe m1,2 Φem1,3 (9) , • Φe m1,3 = diag{nie}k, where i = p − k + 1, ..., p , nie
are the essential orders of the k last outputs of Σm.
• Φe
m1,2is a p − k × k matrix with ∂ciΦ
e
m1,2= nie,
• Φe
m1,1 is a p − k × p − k matrix with: ϕemi,i = sfi,
polynomials ϕe
mi,j = 0 for j > i , and for i < j
the non null polynomials ϕemi,j are such that fj +
1 ≤ ∂minϕem i,j and ∂maxϕem i,j≤ nje− 1,
• the list {nie− fi}p−k = {∆i}p−k, where {∆i}p−k is
the unique minimal list of decoupling indices of Σ, • Φe
m2,3is the m − p × m − p diagonal matrix sσi,
• non null polynomials of Φe m2= Φ
e
m2,1 Φem2,2 are
such that fj+ 1 ≤ ∂minϕemi,j for i ≤ p.
• Note that, for i ≤ p, ∂maxϕem i,j can be greater than
nje.
3. MORGAN’S PROBLEM
3.1 Formulation of the problem and consequences
The static diagonal decoupling without stability of Σ, or Static Morgan’s Problem, can be stated as follows: Given Σ described by (1), does it exists a static state feedback u = F x + Gv = F x + G1v1+ ... + Gpvp, vi ∈ U and
Gi ∈ Rm×1 such that, for any i ∈ 1, 2, ..., p, vi controls
the scalar output yi without affecting the p − 1 other
outputs yj? If such a feedback exists, the transfer matrix
TF G(s) of the closed loop system Σ(C, A + BF, BG) is
TF G(s) = diag{s−r1, ..., s−rp}, ri ∈ R. If G is regular,
(square and invertible) this problem has a solution if and only if the ordered lists {n0i}pand {nie}pare the same. So,
the interactor of Σ is diagonal (Commault et al. [1986]). If {n0i}p 6= {nie}p, Σ will be decouplable if and only if it
is possible to increase the structure at infinity so that it matches to (new) essential orders. This modification can only be achieved by a non regular feedback, that is with a loss of inputs. With such controls, the list of essential orders is not always the minimal infinite structure to reach for the static decoupling.
Static Restricted Morgan’s Problem, (SRMP), is the par-ticular case of Static Morgan’s Problem where the essential orders of Σ should be the infinite structure of the decou-pled system. SRMP is not without interest because, if static decoupling is only possible with a infinite structure ”bigger” than the original essential orders, Σ can always be decoupled without modification of these initial essential orders by a non regular dynamic state feedback. More de-tails on that analysis can be found in Herrera H and Lafay [1993]. Finally, as any non regular decoupling reduces to increase infinite structure, the solution will be based on a theorem of Loiseau [1988] on modifying this structure via non-regular state feedbacks that we remind now:
Theorem 1. Let a linear system for which {n0i} is the infinite structure and {σi} the I2 Morse’s list. Let {pi} a
given list of integers. Note {vi}, {αi} and {πi} the dual
lists of, respectively, {n0i}, {σi} and {pi}. Let {Γi} the
list obtained by arranging the differences (πi− vi) in a
non increasing order. Then, there exists a static state feedback such that the structure at infinity of the closed loop system is the list {pi} if and only if:
v1− vi≥ π1− πi, ∀i ≥ 1, (10) j X i=1 αi≥ j X i=1 Γi, ∀j ≥ 1. (11)
Before addressing the static problem, let us consider now the dynamic solution of decoupling:
3.2 The Dynamic Morgan’s Problem
Proposition 3. (Dion and Commault [1988]). The Dy-namic Morgan’s Problem is solvable if and only if Σ is right invertible and m − p ≥ p − k, k being the rank at infinity of the proper part of the interactor. Essential orders can always be not modified and such minimal solutions need Pp i=1nie− Pp i=1n 0 i integrators.
The dynamic solution consists in bringing Σ in the form ΣΠ (Proposition 2). The list {δi}p−k = {nie− fi}p−k
is not necessarily the list of minimal decoupling indices. Dynamic decoupling amounts to annihilating the indices {δi}p−k applying the following iterative procedure for i =
1, 2, ..., p − k:
- for i = 1, u1 is replaced by an external chain of δ1
integrators controlled by an entry of R∗, for instance v1 = up+1. This external chain is independant of the
(m − p) chains of σi integrators of R∗. This substitution
amounts to multiply the first row of (4) by sδ1. So ∂ϕ
Π1,1
becomes n1eand it is possible, by a left biproper operation,
to eliminate all the other polynomials of the first column of ΦΠ.
- we successively made the same operation for the p − k − 1 other rows of ΦΠ,1, taking at each step a different entry vi
of R∗. This is possible because m − p ≥ p − k.
The final interactor is the p × p matrix diag{snie} and the
system with entries {v1, v2, ..., vp−k, up−k+1, ..., up} is
reg-ularly decouplable. So we need p−k chains of independant integrators coming from a dynamic extension of dimension Pp
i=1nie−
Pp
i=1n 0
i. This dynamic procedure only require
that the number of independent entries of R∗ is at least p − k. The global number of integrators wich are necessary to decouple does not depend of the permutation Π but, even if the list of decoupling indices depends of it. This remark point out one of the main difficulties of Morgan’s problem. Another difficulty is the possible existence of couplings between R∗ and the rest of the system that are not taking into account by the dynamic solution.
4. THE SOLUTION OF SRMP
Let Σm(Cm, Am, Bm) as described in subsection 2.3. We
note Bm = Bb Bs Br ,where Bb = [b1, ..., bp−k],Bs =
[bp−k+1, ..., bp] and Br = B ∩ R∗ = [bp+1, ..., bm]. The
associated entries are u(t) = ub(t) uS(t) ur(t) T
. Define the m × m polynomial matrix ΦS,e
m [s] by: ΦS,em [s] =Φ S,e m1[s] (0) ΦS,em2[s] ΦS,em3[s] , where (12) • ΦS,em1[s] = diag{snie} p • ΦS,em3[s] = diag{sσi }m−p
• ΦS,em2[s] is deduced from the m − p × p matrix Φem2[s] = Φe
m2,1 Φem2,2 by eliminating, in each column i =
1, ..., p, all monomials of degree less than or equal to nieby a left biproper transformation.
• ΦS,e
m [s] has the structure of an interactor.
4.1 Shifted system
Definition 3. The shifted system ΣSm(CmS, ASm, BmS)
asso-ciated with Σm is the invertible system of which ΦS,em [s],
equation (12), is the extended interactor.
Let us extend the state space X of Σm by adding
an extermal dynamic state extension Xa, of dimension
na = Pp−ki=1 ∆i, composed of p − k independant
con-trollable and observable chains of integrators of lenghts ∆1, ..., ∆p−kwith entries w1(t), ..., wp−k(t) in Ua and
out-puts z1(t), ..., zp−k(t) in Ya. Noting ⊕ the external direct
sum of subspaces, the state, control and output spaces of Σmcompleted with this dynamic extension are XS = X ⊕
Xa, US= U ⊕ Ua and YS = Y ⊕ Ya.
The shifted system is obtained by replacing the entries ub(t) of Bb by zi(t) for i = 1, ..., p − k. The state space
realization (CS m, ASm, BmS) of ΣSm is: ASm= Am AS1 . . . ASp−k (0) J∆1 (0) (0) (0) (0) . .. (0) (0) . . . (0) J∆p−k where : (13)
• for i = 1, ..., p − k, J∆i is the upper ∆i× ∆i Jordan
block and the n × ∆i matrices ASiare given by ASi =
bi(n×1) (0)n×∆i−1. • BmS = (0)(n×p−k) Bs(n×k) Br (n×m−p) Ba(na×p−k) (0) (0) , (14)
where Ba= [ba,1, ..., ba,p−k] , ba,i= [0 . . . 0 1] T
being ∆i× 1 vectors.
•
CmS = Cm(p×m) (0)(p×na) . (15)
Then, as in noted in Subsection 3.2 the interactor of ΣS m
is given by
ΦSm[s] = diag{snie}
p (0)p×m−p , (16)
and the corresponding extended interactor of ΣS mis: ΦS,em [s] = diag{s nie} p (0)p×m−p ΦS,em,2 diag{sσi} m−p , (17)
where ΦS,em2[s] is deduced from Φem2[s] by eliminating in
each column j = 1, ...p the monomials of degree less than or equal to nje.
4.2 A convenient formulation of SRMP
SRMP will be solved if it is possible to replace the p − k independant chains of external integrators solving the dynamic Morgan’s problem by p − k independant chains of integrators extracted from Σ. A chain of lenght L is a set of L connected integrators, actived only at its beginning by a function fa(t) to generate a function fe(t),
ie fe(L)(t) = fa(t). Let set of q chains of integrators
defined by fei (Li)
(t) = fai(t), i = 1, ..., q. For SRMP,
we require that each function fi
a(t) contains at least one
input of the the system. These chains are independant if, ∀j 6= i, the term containing the inputs in fi
a(t) is not
a linear combination of terms containing the inputs in fj
a(t), and if fei(t) is not a linear combination of functions
fj
e(t). The following lemma characterize internal chains
that will increase the infinite structure without changing the essential orders: it is a new formulation for SRMP, generalizing in a non trivial way Lemma 4.2 in (Herrera H and Lafay [1993])valid when k = p − 1.
Lemma 1. Let a right invertible system Σm with R∗ =
V∗, k the rank of the proper part of its interactor and
∆i
(p−k) its decouplability indices (say the minimal
decoupling indices of Σ). Then, SRMP has a solution if and only if it is possible to extract from R∗ p − k independant chains of integrators of lengths ∆1, ..., ∆p−k
described by fi e(t)
(∆i)
= fi
a(t), i = 1, ..., p − k such that:
(a) The output fi
e(t) of each chain of integrators is only
function of the vector state space x(t) and these p − k functions are independant,
(b) For i = 1, ..., p − k, the entry fai(t) of each chain of integrators do not contain derivatives of inputs uj(t), j = p − k + 1, ..., p.
(c) For i = 1, ..., p − k, the entry fai(t) of each chain of integrators do not contain derivatives of inputs uj(t) of order greather than or equal to ∆j, for all
j = 1, ..., p − k .
Proof 1. of Lemma 1 IF. Let a state feedback (F, G) which decouples Σm(Cm, Am, Bm). According to Dion
and Commault [1988] this feedback is equivalent to the precompensator C(s) = (Im− F (sIn− Am)−1Bm)−1G =
W1,m(s)
X(s)
, where X(s) is a proper m − p × p matrix and, noting d = s−1, the p × p proper part W1,m(s) of the
interactor of Σm can be written as:
W1,m(s) = d∆1 ˆ ϕ2,1 . .. (0) d∆p−k ( ˆϕi,j) ... 1 .. . (0) . .. ˆ ϕp,1 ϕˆp,p−k 0 · · · 1 . (18) ˆ
ϕi,j[d] are polynomials in d = s−1 such that ∂ ˆϕi,j[d] <
∆j = nje− fj for i ≤ p − k.
Matrix G of the feedback (F, G) is obtained (Herrera H [1992]) by: G = lim s→∞W1,m(s) = (0)p−k×p−k (0) (gi,j)k×p−k Ik where gi,j∈ R .
So, the control law u(t) = F x(t) + Gv(t) is given by: 4
ui(t) = Fix(t), i = 1, ..., p − k (a) Fix(t) +Pp−kj=1gi,jvj(t) + vi(t), i = p − k + 1, .., p (b) (19) As Σm,(F,G)(Cm, Am + BmF, BmG) is assumed to be
decoupled without changing the essential orders, we have:
y(nie) i (t) = ( u(∆i) i (t) = vi(t), i = 1, ...p − k u(0)i (t) = vi(t), i = p − k + 1, ..., p. (20)
Therefore, equations (20) and (19a) describe p − k chains of lenghts ∆1, ..., ∆p−k generating independant functions
Fix(t) of the state x(t). Moreover, from (19b), these
chains are not activated by derivatives of the entries up−k+1(t), ..., up(t) and for i and j = 1, ..., p − k, the ith
chain is not actived by derivatives of vj(t), i 6= j. But, by
equation (19b), if the role of the chains is only to make Σmregularly decouplable, the ith chain can be eventually
activated by derivatives of entries ui(t) of order less than
∆ifor i = 1, ..., p−k, because for each chain must generate
a function of the state x(t), say: fai(t) = fei (∆i)
(t) and fi
a(t) = Fix(t)
According Loiseau [1988], since any increasing of infinite structure can only be done using entries of R∗, entries v1(t), ..., vp−k(t) should contain independant linear
combi-nations of entries of R∗.
(Only if ): Assume that these p − k independant chains can be constructed. For i = 1, ..., p − k, each chain is characterized by its lenght ∆i, its entry fai(t) and its
output fi
e(t) linked by fei (∆i)
(t) = fi
a(t). As fei(t) is
uniquely a function of the state space of Σm, we can write
fei(t) = Fix(t), where Fi ∈ R1×n, and as the chains are
independant, rank ¯F = F1 .. . Fp−k = p − k.
Now, consider the entries of the chains. As these chains are extracted fom R∗, each function fi
a(t) contains at least
one linear combination of the entries up+1(t), ..., um(t) and
these linear combinations are independant: this implies that dim B ∩ R∗ ≥ p − k. This is the necessary and sufficient condition of dynamic decoupling for Σ. For static decoupling, we must add the condition: Pp−k
i=1 ∆i≤
Pm−p
i=1 σi, where {σi}m−pis the Morse’s list I2 .
By assumption, no derivative of entries up−k+1(t), ..., up(t)
appears in functions fi
a(t). However there may be
terms which depend on the state x(t) and/or on entries up−k+1(t), ..., up(t), and/or, for i = 1, ..., p − k, on
deriva-tives of entries ui(t) of order less than ∆i, (because each
function fi
a(t) = fei (∆i)
(t) does not include derivative of ui(t) of order greather than or equal to ∆i from the fact
that fi
e= Fix(t). So, for i = 1, ..., p − k, the general form
of functions fi a(t) is: fai(t) = fa1i (up+1(t), ..., um(t)) + fa2i (x(t), up−k+1(t), ..., up(t)) + p−k X j=1 fa3,ji (u(1)j (t), ..., u(∆j−1) j (t)). (21)
fa1i are nonzero and independant functions. As u(j)i (t) = y(fi+j)
i (t) = y
(nie−∆i+j)
i (t), the Laplace’s
transform of each function fai(t) is:
fai(s) = fa1i (up+1(s), ..., um(s)) + fa2i (x(s), up−k+1(s), ..., up(s)) + p−k X j=1 Ψi,j[s]yj(s), (22)
where the degree of each polynomial Ψi,j[s] is less than
nje.
We can now define the non regular feedback u(t) = F x(t)+ Gv(t) by: F = ¯ Fp−k×n 0k×n F0 m−p×n and G = 0p−k×k 0p−k×p−k Ik 0k×p−k 0 m−p×k G0m−p×p−k .
Here rank ¯F = p − k and rank G0= p − k ≤ m − p.
The possibly non regular feedback (F0, G0) acts only on
R∗ to create the p − k independant chains of integrators.
Noting Br = ImB ∩ R∗ and xR∗(t) trajectories in R∗,
this part of feedback is such that: F0xR∗(t)+BrG0v∗(t) =
f1 a1(•) .. . fa1p−k(•) , where v∗(t) = vp−k+1(t) .. . vp(t) . Then, as y(fi) i (t) = ui(t) = fei(t) for i = 1, ..., p − k, we obtain: L(y(nie) i (t)) = feix(s)s∆i = fai(t) = fa1i (•) + fa2i (•) + Pp−k
j=1Ψi,j[s]yj(s) with ∂Ψi,j[s] < nje. Then,
outputs y(s) and new entries v(s) are linked by: H1,1[s] (0)p−k×k H2,1[s] H2,2[s] y(s) = V (s) = V1(s) V2(s) , (23) where
1- The p − k × p − k polynomial matrix H1,1is given by:
H1,1 = sn1e− Ψ 1,1 . .. (−Ψ i,j) (ϕe mi,j− Ψi,j) . .. sn(p−k)e− Ψ p−k,p−k .
2- H2,1=ϕemi,j[s] is a k × p − k polynomial matrix
3- H2,2[s] = diag{sn(p−k+1)e, ..., snpe}k. 4- V1(s) = fa11 (•) + fa21 (•) .. . fa1p−k(•) + fa2p−k(•) and V2(s) = vp−k+1(s) = up−k+1(s) .. . vp(s) = up(s) .
As the polynomial of highest degree of each column of H[s] is on the diagonal the matrix, the infinite structure coin-cides with essential orders. This system is then regularly decouplable without increasing the essential orders.This ends the proof of Lemma 1.
Remark 2. Functions fi
a(t) cannot contain derivatives ou
entries of R∗. If that were the case, for exemple for the
first chain, the effective lenght of this chain would be less than ∆1. Indeed, suppose that this chain of lenght ∆1 is
activated by up+1(t) and by u (1) p+j(t), for j = 1, ..., m − p. Then f1 e (∆1)(t) = f1 a(t) = up+1(t) + u (1) p+j(t) and 5
f1 e (∆1−1 )(t) = R f1 a(t)dt = R up+1(t)dt +R u (1) p+j(t)dt =
f (x(t)) + up+j(t). The effective lenght of the chain is
∆1− 1.
Remark 3. Inputs u1(t), ..., up−k(t) are suppressed, while
inputs up−k+1(t), ..., up(t) are preserved.
Remark 4. If functions fi
a(t) do not include derivatives of
inputs ui(t), i = 1, ..., p − k, the system is decoupled.
Lemma 1 will help us to derive conditions on Σmfor ensure
that such p − k independant decoupling chains exist. To characterize the maximal lenghts of the chains of R∗ which are not actived by derivatives of entries uj(t) of
order higher or equal to ∆j , j = 1, ..., p, it is possible to
apply Theorem 4.1 in Herrera H and Lafay [1993] taking into account the following remark:
Remark 5. The chains of R∗ satisfying items (b) and (c) of Lemma1 for Σm are the same as the chains of R∗
which are not actived by derivatives of entries w1, ..., wp−k
up−k+1, ..., up for ΣSm.
Theorem 4.1 in Herrera H and Lafay [1993] becomes: Proposition 4. Let ΣS
m be given. Let {ci}m the
con-trollability indices of (ASm, BSm). Then the set {αi}m−p
of maximal lenghts of subchains of R∗ not activated by derivatives of inputs w1, ..., wp−k, up−k+1, ..., upis
αi= ci, for i = p + 1, ..., m. (24)
4.3 Main result: the solution of SRMP
Theorem 2. Let the right invertible system Σ be given with R∗ = V∗ and k the rank at infinity of the proper part of its interactor. Let Σm deducted by regular
state feedback from Σ such that the infinite structure {∆i}(p−k)of the proper part of the interactor of Σmis the
minimal list of decoupling indices of Σ. Let {αi}(m−p) the
controllability indices of the entries of R∗ for the shifted
system ΣS
massociated wit Σm. Then SRMP has a solution
if and only if, for all i ≥ 1,
i X j=1 ˆ αj≥ i X j=1 γj, (25)
where { ˆαi}sup αi the dual list of {αi}(m−p) and {γi}sup ∆i
is the dual list of {∆i}(p−k).
Proof 2. Sufficiency:
Theorem 1 cannot be applied directly because it trans-forms globally a list {n0i}p−k into a list {nie}p−k, and for
decoupling, the transformations will be done done term by term. If this is was not the case, the essential orders would be changed. But, as mentionned in Lafay [2013] it suffices to apply Theorem 1 choosing for list {n0i}p−k the list {1}p−k and try to turn it into the list {1 + ∆i}p−k.
This amounts to build, from the m−p chains of integrators of lengths σi of R∗, p − k independent chains of lengths
{∆i}. Note that condition (10) is still always true. So
there remains only conditions (11).
Let us now return to SRMP. It is sufficient to replace the dynamic extension of Subsection 4.1 by p − k independant chains of lenghts {∆i}(p−k) extracted from R
∗ and
satis-fying Lemma 1. This is equivalent to create, from ΣSm,
p − k independant chains extracted from R∗ not actived by derivatives of entries w1, ..., wp−k, up−k+1, ..., up(cf
Re-mark 5). By Proposition 4, such chains should be built from the maximal subchains of R∗ which are not undivid-ually actived by derivatives of w1, ..., wp−k, up−k+1, ..., up.
According to the construction of the shifted system, and Remark 5, integers {αi}m−p represent equivalently the
maximal lenghts of subchains of R∗ not activated by derivatives of inputs up−k+1, ..., up and for i = 1, ..., p − k
not activated by derivatives of inputs ui of order greather
than or equal to ∆i. Applying Proposition 1 as
men-tionned at the beginning of this proof, we obtain condition (26) which is sufficient to solve SRMP.
Necessity: The necessity of this condition comes from two facts. First: by Proposition 2, if there is a solution to SRMP, there are some which which require only (p-k) increases of infinite structure. Conditions (11 there-fore requires that the minimum list of decoupling indices contain only (p-k)termes. Secondly: the list of minimal decoupling indices contains (p-k) terms and ”minore” all the other lists of decoupling indices. In other words, if there is no solution for these indices, so with the order of outputs of Σm, it does not exist solution for any other list
if decoupling indices coming from other permutations of outputs of Σ. This ends the proof .
Corollary 1. The condition (26) of Theorem 2 is a nec-essary and sufficient condition for the general Morgan’s problem for a system Σ when m − p = k.
5. COMPARISONS
At our knowledge, the more advanced structural results on static Morgan’s problem are in Herrera H and Lafay [1993], Zagalak et al. [1998] and Lafay [2013].
5.1 Herrera H and Lafay [1993]
In this paper, SRMP is solved when k = p−1. The solution was stated as follows, with the notations adopted here: Theorem 3. Let the right invertible system Σm be
given with R∗=V∗. Suppose k = p − 1. Let δ 1 be
the (nonzero)infinite structure of the interactor Π1m[s],
and {αr,i}m−p the controllability indices of the pair
(Am, [Bs|Br]) related with the columns of Br. Then the
SRMP has a solution if and only if:
m−p
X
j=1
αr,j≥ δ1. (26)
Le Σm = (Cm, Am, Bm) and note Bm = [b1|Bs|Br] as in
Section 4. In this particular case, the list of decoupling indices contains only one term δ1 =Ppi=1nie−Ppi=1n0i.
Then this list is minimal. A static solution will exist if and only if the sum of lenghts of the subchains of R∗ not actived by derivatives of entries (u2, ..., up) is greather than
or equal to δ1 which is strictly less than n1e. Consider
now the shifted system associated with Σm. Following
equations (13) and (14), we have:
ASm= Am A S 1 (0) J∆1 where (27) 6
• Jδ1 is the upper δ1 × ∆1 Jordan bloc, and matrix ASiare given by AS1 = b1(n×1) (0)n×∆1−1. • BS m1 = (0)(n×1) Bs(n×p−1) Br (n×m−p) ba(δ1×1) (0) (0) with (28) ba = [0 . . . 0 1] T .
Let us compare the controllability indices {αi}m−1 of the
the pair (Am, [Bs|Br]) and the controllability cSi
m of
ΣS
m. Using the Brunovsky’s procedure, Brunovski [1970],
we have cS
1≥ n1e, then
• if αi≤ n1e, then αi= cSi+1, and
• if αi> n1e, we can have cSi+1< αi but cSi+1> n1e.
Then as δ1 ≤ n1e the two Theorems are equivalents for
the existence of a solution for SRMP.
5.2 Zagalak et al. [1998]
In this paper, the authors consider systems Σ with special conditions on dimensions as m = 2p and Pp
i=1δi =
Pp
i=1σi, and mainly without couplings between R ∗ and
the blocks of infinite structure, ie Φem2= Φem2,1 Φem2,2=[0]
in the extended interactor of Σm. The problem was
however difficult althought these assumptions seem very simplistic. Certainly, the authors propose factorizations which must have a link with the minimal list of decoupling indices but, unless I have not properly understood their approach, this structural information does never appear explicitly.
6. CONCLUSION
In this paper, we propose a necessary and sufficient con-ditions for the row by row decoupling problem without modifying the essential orders. Even in this special form, the problem was recognized as structurally difficult. It remains to solve the general problem which is much more difficult because we do not know yet how to define minimal structures for the decoupled system.
Appendix A. MINIMAL LIST OF DECOUPLING INDICES
We give here the procedure to obtain the minimal list {∆i}p−k;
Let the interactor Φ[s] = Φ1[s] (0) Φ2[s] Φ3[s]
= [ϕi,j[s]] of Σ
that is supposed to be in the form (4) .
For columns 1, 2, ..., p−k, we note piethe maximum degree
of the ith column of Φ1[s] , γi = nie− pie and define the
index ∆ = min {γ}p−k. We will first prove that ∆ does not depend on the permutation Π such as the interactor of ΣΠis under the form (4) and in a second step that it is
the smallest decoupling index possible for ΣΠ.
According to Proposition 2, the polynomial of highest degree of each column of ΦΦ1[s]
2[s]
belongs to Φ2[s]. Let
ϕt,j[s], with j ≤ p − k and t > p − k, this polynomial for
the jth column of Φ[s]. Its degree is nje. Let Π−1 the
permutation of the tth and jth columns of Φ[s], which corresponds to the permutation Π of the tth and jth outputs of (1). Φ[s]Π−1 is no longer an interactor. A.1 ∆ is not modified
To simplify the notations and without loss of generality, we assume that j = 1 and t = p. Noting (only for the next equation) α = p − k, Φ[s]Π−1 is given by:
0 0 . . . . 0 ϕ1,1 0 ϕ2,2 . . . . 0 ϕ2,1 . . . . 0 ϕα,2 . ϕα,α . . 0 . 0 ϕα+1,2 . . ϕα+1,α+1 . . . . . . . 0 . . . 0 ϕp−1,2 . . 0 0 ϕp−1,p−1 . ϕp,p ϕp,2 . ϕp,α 0 0 0 ϕp,1 (A.1)
We will determine a biproper matrix BΠ(s) such that
ΦΠ[s] = BΠ(s)Φ[s]Π−1 is the interactor of ΣΠ. First we
cancel by a left biproper operation B1(s) the polynomials
ϕj,1, j = 1, ..., p − 1, of the pth column of (A.1). B1(s)
always exists since ∂ϕp,1[s] = n1e≥ ∂ϕj,1[s], ∀j. Choosing
B1(s) = 1 0 . . . . 0 −ϕ1,1 ϕp,1 0 1 0 . . . 0 −ϕ2,1 ϕp,1 . . . . . 0 . . . . 0 1 −ϕp−1,1 ϕp,1 0 0 . . . . 0 1 , (A.2) B1(s)ΦΠ−1= ˆ Φ1(s) (0) ˆ Φ2(s) ˆΦ3[s] = [ ˆϕi,j] , (A.3)
where ˆΦ1[s] is a p − k × p − k matrix, and where ˆΦ3[s] and
Φ3[s] differ only by ˆϕp,p= ϕp,1while1 ϕˆp,1 = snpe=ϕp,p.
Except this permutation, the other polynomials of the pth row of B1(s)ΦΠ−1 are not modified. Note that ˆΦ1(s)
and ˆΦ2(s) are no longer necessarily polynomial matrices.
Consider now the other columns of B1(s)Φ[s]Π−1:
Lemma 2. For j = 2, ..., p − k, the highest degree the jth columns of B1(s)ΦΠ−1 and Φ[s] is not changed and the
polynomials of degree nje are in ˆΦ2[s].
Proof 3. The degrees of columns are the same because B1(s) is biproper. In equation (A.3), ϕi,jis transformed as
ˆ
ϕi,j=ϕi,j -ϕi,1
ϕp,1ϕp,j. For i and j = 1, ..., p − k, ∂ϕi,j< nje.
As ∂ϕi,1 < ∂ϕp,1 and ∂ϕp,j ≤ nje, we have ∂ϕϕi,1
p,1ϕp,j <
nje, which ends the proof of Lemma 2.
We will now prove the following Lemma:
Lemma 3. : ∆ is the same for Φ[s] and for B1(s)ΦΠ−1.
Proof 4. Let B1(s)ΦΠ−1 and nie− ∂ ˆϕi,j = ˆγi, for i =
1, ..., p − k
1- In (A.3), each polynomial ˆϕi,1 of ˆΦ1[s] is given by:
ˆ
ϕi,1= −ϕp,p
ϕi,1
ϕp,1
, (A.4)
with ∂ϕp,p = npe and ∂ϕp,1 = n1e. Then npe− ∂ ˆϕi,j =
n1e− ∂ϕi,j. So ˆγ1 = γ1. Especially, if we had γ1 = ∆,
1 We could always suppose that ϕ
p,1is normalized as sn1e
this index is transmitted in the first column of B1(s)ΦΠ−1.
But still nothing proves here that ∆ = min {ˆγi}p−k.
2 - For the other columns of ˆΦ1[s], ˆϕi,j = ϕi,j− ϕp,j ϕi,1
ϕp,1,
and then: ∂ ˆϕi,j≤ max (∂ϕi,j, ∂ϕp,j ϕi,1
ϕp,1).
Then or ˆγj = mini{nje− ∂ϕi,j}p−k = γj, or ˆγj =
mini{nje− ∂ϕp,j− ∂ϕi,1+ ∂ϕp,1}p−k: in this last case,
as ∂ϕp,1− ∂ϕi,1≥ γ1, we have:
nje− ∂ ˆϕi,j≥ nje− ∂ϕp,j+ γ1= γ1+ c, where c ≥ 0.
So, ˆγj ≥ min {γj, γ1+ c}. If γj = ∆ = γ1, there may
be cancellation of the terms of highest degree of the polynomial and then ˆγj > ∆. But, in this case, ∆ is
still in the first column, as we have seen in item 1. The consequence is that, for each column j,
ˆ
γj ≥ min (γj, γ1) (A.5)
and ∆ = min {ˆγi}p−k, which ends the proof of Lemma 3.
A.2 ∆ is the smallest decoupling index of Σ
Suppose that, for the rth column of B1(s)ΦΠ−1, nre−
pre= ∆ and ∂ ˆϕs,r= pre. Permute (by Πr−1) the rth and
the (p − k)th columns of B1(s)Φ[s]Π−1. We obtain
B1(s)ΦΠ−1Πr−1= ˆ Φ1,r(s) (0) ˆ Φ2,r(s) ˆΦ3[s] = ˆϕr i,j (A.6)
There exists a biproper transformation B2(s) such that
the interactor of the new permuted system is given by Φr[s]=B2(s)B1(s)Φ[s]Π−1Πr−1 = ˆϕr i,j. From (A.6),
B2(s) has the following structure:
B2(s) =B2,1
(s) (0) B2,2(s) Ik
, (A.7)
where B2,1(s) is a biproper p − k × p − k matrix such
that B2,1(s) ˆΦ1,r(s)Πr−1 is structured as an interactor.
As B2(s) and B2,1(s) are biproper, the maximum
de-grees of each column j of B2(s)B1(s)ΦΠ−1Πr−1 and of
B2,1(s) ˆΦ1,r[s] are respectively equal to nje and pie. Then
ˆ
ϕr p−k,p−k = spe, ˆϕr i,p−k = 0 for i = 1, ..., p − k − 1, and the monomials of degree less than or equal to pe have been canceled by B2,2(s) in the polynomials ˆϕr i,p−k = 0
for i = p − k + 1, ..., p . So, the decoupling index of the new (p − k)th output, noted ∆p−k is ∆p−k= ∆. It is the
smallest possible for SRMP from Lemmas 1 and 3. Remark 6. From (A.5) and as B2,1(s) is biproper, the new
quantities ˆγr icomputed for the interactor Φr[s] are greater
than or equal to ∆p−k= ∆.
A.3 the minimal list of decoupling indices
Let Φr[s] defined in subsection A.2. Note
˜ Φ[s) = ˜ Φ1[s] (0) ˜ Φ2[s] ˜Φ3[s] = [ ˜ϕi,j[s]] . (A.8)
where ˜Φ1[s] has the structure of a p − k − 1 × p − k − 1
interactor. We apply the same procedure as before with regard to the columns 1, 2, ..., p − k + 1 of Φr: we obtain
a second decoupling index ∆p−k−1 ≥ ∆p−k which is as
small as possible at this step. This procedure is iterated until obtaining the unique list {∆i}(p−k)=∆1 ≥ ∆2 ≥
... ≥ ∆p−k = ∆ of the p − k integers. The minimality
comes from the fact that at each iteration i, the increase of infinite structure ∆−i is the smallest that we can obtain by permutation of the outputs of Σ. Then this list ”minore” all the other lists at the sense of Definition 2.
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